Finite Group Databasebeta

← #364670

Subgroups — #364670

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#13872C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10721C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×C8×D4—24no—
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2487—24no—
25#2487C418yescomplemented
26C2×#400—21no—
27C2×#399—21no—
28C2×#389C2212yes—
29C2×#386—21no—
30C2×#386—21no—
31C2×#385—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#1824—24no—
37C2×#340—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1441—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8—42no—
64C23×C8D414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4C2×C414yes—
74C2×C4×D4D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2×C414yes—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4D411yescharacteristic
98C22×C22⋊C4D411yescharacteristic
99C22×C42C2311yescharacteristic
100C8×D4—232no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8D414yes—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113#400—28no—
114#399—28no—
115#389—24no—
116#389C2×C418yescomplemented
117#386—28no—
118#386—44no—
119#385—28no—
120#380—22no—
121#380—24no—
122#380—22no—
123#380—22no—
124#380—24no—
125#380—42no—
126#380C2×C414yes—
127#380C2×C418yescomplemented
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—42no—
142#340—42no—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C22⋊C414yes—
150C22×D4SD1614yes—
151C22×D4Q1614yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—28no—
195C22×C8—28no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—28no—
209C22×C8—44no—
210C22×C8—48no—
211C22×C8—48no—
212C22×C8—48no—
213C22×C8—48no—
214C22×C8—44no—
215C22×C8—44no—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8#4114yes—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22⋊C414yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4#4114yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4C22⋊C414yes—
284C2×C22⋊C4C22⋊C414yes—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C22×C412yes—
290C2×C42C2×D412yes—
291C4⋊C8—28no—
292C22⋊C8—28no—
293C22⋊C8—28no—
294C4×C8—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—82no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24C2×C22⋊C412yes—
315C24C2×C22⋊C412yes—
316C24#12112yes—
317C24#12112yes—
318C24C2×SD1612yes—
319C24C2×SD1612yes—
320C24C2×Q1612yes—
321C24C2×Q1612yes—
322C24#13512yes—
323C24#13512yes—
324C24C22×D411yescharacteristic
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4#103116yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—88no—
415C22×C4—84no—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C4×D414yes—
421C22×C4C4×D414yes—
422C22×C4#12014yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4#12314yes—
426C22×C4C2×SD1612yes—
427C22×C4C2×Q1612yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—416no—
450C2×C8—416no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8#12314yes—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—88no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—84no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23#38614yes—
529C23C2×#10314yes—
530C23C2×#10314yes—
531C23#41614yes—
532C23#46514yes—
533C23#46514yes—
534C23#46714yes—
535C23#46714yes—
536C23C22×C22⋊C411yescharacteristic
537C23C2×#12111yescharacteristic
538C23C2×#12111yescharacteristic
539C23#54612yes—
540C23#54712yes—
541C23C22×SD1611yescharacteristic
542C23C22×Q1611yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×#13511yescharacteristic
545C23#57612yes—
546C23#57712yes—
547D4—232no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—816no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4#38614yes—
611C2×C4C2×#10314yes—
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C4×#4112yes—
615C2×C4#52012yes—
616C2×C4C2×#12011yescharacteristic
617C2×C4C2×#12311yescharacteristic
618C2×C4C2×#12311yescharacteristic
619C2×C4#52511yescharacteristic
620C2×C4#52912yes—
621C2×C4#53411yescharacteristic
622C2×C4#53611yescharacteristic
623C2×C4#53912yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4#54712yes—
627C2×C4#54712yes—
628C2×C4#55112yes—
629C2×C4#55212yes—
630C8—24no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—28no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22#189218yes—
656C22C2×#38611yescharacteristic
657C22#233912yes—
658C22#242512yes—
659C22#248611yescharacteristic
660C22#253411yescharacteristic
661C22C22×#10311yescharacteristic
662C22C2×#41611yescharacteristic
663C22#289712yes—
664C22C2×#46511yescharacteristic
665C22C2×#46511yescharacteristic
666C22C2×#46711yescharacteristic
667C22C2×#46711yescharacteristic
668C22#305711yescharacteristic
669C22#305811yescharacteristic
670C22#306212yes—
671C22#329512yes—
672C22#329712yes—
673C22#329912yes—
674C22#330112yes—
675C22#333512yes—
676C22#333712yes—
677C22#333812yes—
678C22#333912yes—
679C4—24no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—84no—
684C4—84no—
685C4—84no—
686C4—162no—
687C4C4×#12311yescharacteristic
688C4#232211yescharacteristic
689C4#235912yes—
690C4#240511yescharacteristic
691C4#240611yescharacteristic
692C4#241912yes—
693C2—28no—
694C2—44no—
695C2#1396811yescharacteristic
696C2C2×#189211yescharacteristic
697C2#2097511yescharacteristic
698C2#2248712yes—
699C2#2248812yes—
700C2#2523412yes—
701C2#2523812yes—
702C2#2765811yescharacteristic
703C2#2766211yescharacteristic
704C2#2809511yescharacteristic
705C2#2809711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1824C2semidirect
4. C22×#340C2semidirect
6. #11228C2semidirect
7. #11200C2semidirect
8. #10880C2semidirect
9. #10730C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect
25. #2487C4semidirect
116. #389C2×C4semidirect
127. #380C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C8×D42
C2×#24872
C2×#18242
C22×#3402
#138722
#112282
#112002
#108802
#107302
#107212